Integral equations and representations of some Hermite-type families of orthogonal matrix polynomials

Manuel Domínguez de la Iglesia Integral equations and representations of some Hermite-type families of orthogonal matrix polynomials Departamento de Análisis Matemático, Universidad de Sevilla ORTHONET 2013 Logroño, February 23, 2013 1 sábado 23 de febrero de 13 Manuel Domínguez de la Iglesia Integral equations and representations of ONE Hermite-type family of orthogonal matrix polynomials Departamento de Análisis Matemático, Universidad de Sevilla ORTHONET 2013 Logroño, February 23, 2013 1 sábado 23 de febrero de 13 HERMITE POLYNOMIALS Consider the classical Hermite polynomials Hn(x)=(1)nex2ex2(n) satisfying ZR Hn(x)Hm(x)ex2dx =2 nn!p⇡nm 2 sábado 23 de febrero de 13 HERMITE POLYNOMIALS 1810 1859 Consider the classical Hermite polynomials Hn(x)=(1)nex2ex2(n) satisfying ZR Hn(x)Hm(x)ex2dx =2 nn!p⇡nm 2 sábado 23 de febrero de 13 HERMITE POLYNOMIALS Consider the classical Hermite polynomials Hn(x)=(1)nex2ex2(n) satisfying ZR Hn(x)Hm(x)ex2dx =2 nn!p⇡nm 2 sábado 23 de febrero de 13 HERMITE OR WAVE FUNCTIONS 3 sábado 23 de febrero de 13 HERMITE OR WAVE FUNCTIONS The Hermite or wave functions are defined by n(x)= 1 p2nn!p⇡ex2/2Hn(x) 3 sábado 23 de febrero de 13 HERMITE OR WAVE FUNCTIONS The Hermite or wave functions are defined by n(x)= 1 p2nn!p⇡ex2/2Hn(x) ( n)nis a complete orthonormal set in L2(R), i.e. ZR n(x) ⇤ m(x)dx =nm 3 sábado 23 de febrero de 13 HERMITE OR WAVE FUNCTIONS The Hermite or wave functions are defined by n(x)= 1 p2nn!p⇡ex2/2Hn(x) Let D:L2(R)!L2(R)theSchr¨odinger operator with quadratic potential [Df](x)= d2 dx2f(x)x2f(x) and F:L2(R)!L2(R) the continuous Fourier transform [Ff](x)= 1 p2⇡ZR f(t)eixtdt 3 sábado 23 de febrero de 13 HERMITE POLYNOMIALS Some consequences for the Hermite polynomials: H00 n(x)2xH0 n(x)+2nHn(x)=0 Hn+1(x)2xHn(x)+2nHn1(x)=0 5 sábado 23 de febrero de 13 HERMITE POLYNOMIALS Some consequences for the Hermite polynomials: H00 n(x)2xH0 n(x)+2nHn(x)=0 Hn+1(x)2xHn(x)+2nHn1(x)=0 5 sábado 23 de febrero de 13 HERMITE POLYNOMIALS Some consequences for the Hermite polynomials: H00 n(x)2xH0 n(x)+2nHn(x)=0 Hn+1(x)2xHn(x)+2nHn1(x)=0 From the integral equation we get ex2/2H2n(x)=(1)nr2 ⇡Z1 0 et2/2H2n(t) cos(xt)dt ex2/2H2n+1(x)=(1)nr2 ⇡Z1 0 et2/2H2n+1(t)sin(xt)dt 5 sábado 23 de febrero de 13 HERMITE POLYNOMIALS Some consequences for the Hermite polynomials: H00 n(x)2xH0 n(x)+2nHn(x)=0 Hn+1(x)2xHn(x)+2nHn1(x)=0 From the integral equation we get ex2/2H2n(x)=(1)nr2 ⇡Z1 0 et2/2H2n(t) cos(xt)dt ex2/2H2n+1(x)=(1)nr2 ⇡Z1 0 et2/2H2n+1(t)sin(xt)dt Additionally, using the Laplace and the Fourier transform on the di↵erential equation we obtain the following integral representations of (Hn)n Hn(x)= n! 2⇡iI ez2+2zx dz zn+1 Hn(x)=(2i)nex2 p⇡ZR eu2+2ixuundu 5 sábado 23 de febrero de 13 CHRISTOFFEL-DARBOUX KERNEL The Christo↵el-Darboux (CD) kernel Kn(x, y)= n1 X k=0 k(x) k(y) describes the statistical properties of the eigenvalues of a random matrix M in the space of (n⇥n) Hermitian matrices with the measure µ(M)=e Tr(M2)dM(GUE, Mehta). 6 sábado 23 de febrero de 13 CHRISTOFFEL-DARBOUX KERNEL The Christo↵el-Darboux (CD) kernel Kn(x, y)= n1 X k=0 k(x) k(y) describes the statistical properties of the eigenvalues of a random matrix M in the space of (n⇥n) Hermitian matrices with the measure µ(M)=e Tr(M2)dM(GUE, Mehta). The last particle distribution is given by the Fredholm determinant F(s)=P[max s]=det(IsKn) where sis the indicator function of the interval [s, 1) and Kn:L2(R)!L2(R)istheintegral operator [Knf](x)=ZR Kn(x, y)f(y)dy 8f2L2(R) 6 sábado 23 de febrero de 13 CHRISTOFFEL-DARBOUX KERNEL The Christo↵el-Darboux (CD) kernel Kn(x, y)= n1 X k=0 k(x) k(y) describes the statistical properties of the eigenvalues of a random matrix M in the space of (n⇥n) Hermitian matrices with the measure µ(M)=e Tr(M2)dM(GUE, Mehta). The last particle distribution is given by the Fredholm determinant F(s)=P[max s]=det(IsKn) where sis the indicator function of the interval [s, 1) and Kn:L2(R)!L2(R)istheintegral operator [Knf](x)=ZR Kn(x, y)f(y)dy 8f2L2(R) THEOREM (Tracy-Widom, 1994) The log derivative of the Fredholm determinant R(s)=@slog(det(IsKn)) solves the sigma-form of the PIV equation (R00)2+ 4(R0)2(R0+2n)4(sR0 R)2=0 6 sábado 23 de febrero de 13 MATRIX VALUED FUNCTIONS Let Wbe a weight matrix (positive definite and finite moments). Consider L2 W(R,CN⇥N)theweighted space with the inner product hF,GiW=ZR F(x)W(x)G⇤(x)dx 7 sábado 23 de febrero de 13 MATRIX VALUED FUNCTIONS Let Wbe a weight matrix (positive definite and finite moments). Consider L2 W(R,CN⇥N)theweighted space with the inner product hF,GiW=ZR F(x)W(x)G⇤(x)dx Asequence(Pn)nof matrix orthonormal polynomials (MOP) with respect to Wis a sequence satisfying deg Pn=n, hPn,PmiW=INnm 7 sábado 23 de febrero de 13 MATRIX VALUED FUNCTIONS Let Wbe a weight matrix (positive definite and finite moments). Consider L2 W(R,CN⇥N)theweighted space with the inner product hF,GiW=ZR F(x)W(x)G⇤(x)dx Asequence(Pn)nof matrix orthonormal polynomials (MOP) with respect to Wis a sequence satisfying deg Pn=n, hPn,PmiW=INnm They were studied for the first time by M. G. Krein (1949) 7 sábado 23 de febrero de 13 SOME NOTATION AN= 0 B B B B B @ 0⌫10··· 0 00⌫2··· 0 . . . . . . . . ..... . . 00 0··· ⌫N1 00 0··· 0 1 C C C C C A ,⌫ i2R,JN= 0 B B B @ N1··· 00 . . ..... . . . . . 0··· 10 0··· 00 1 C C C A zJ= 0 B B B @ zN1 ... z 1 1 C C C A ,z J= 0 B B B B B @ 1 zN1 ... 1 z 1 1 C C C C C A 8 sábado 23 de febrero de 13 THE EXAMPLE 9 sábado 23 de febrero de 13 THE EXAMPLE 9 sábado 23 de febrero de 13 THE EXAMPLE 9 sábado 23 de febrero de 13 THE EXAMPLE Let us consider L2 W(R,CN⇥N) with the weight matrix W(x)=e x2eAxeA⇤x,x2R and the family of MOP (Pn)nsatisfying P00 n(x)+P0 n(x)(2xI+2A)+Pn(x)(A22J)=(2nI2J)Pn(x) 10 sábado 23 de febrero de 13 THE EXAMPLE Let us consider L2 W(R,CN⇥N) with the weight matrix W(x)=e x2eAxeA⇤x,x2R and the family of MOP (Pn)nsatisfying P00 n(x)+P0 n(x)(2xI+2A)+Pn(x)(A22J)=(2nI2J)Pn(x) The orthogonal matrix-valued functions (complete in L2(R,CN⇥N)) n(x)=e x2/2Pn(x)eAx are simultaneously eigenfunctions 10 sábado 23 de febrero de 13 THE EXAMPLE Let us consider L2 W(R,CN⇥N) with the weight matrix W(x)=e x2eAxeA⇤x,x2R and the family of MOP (Pn)nsatisfying P00 n(x)+P0 n(x)(2xI+2A)+Pn(x)(A22J)=(2nI2J)Pn(x) 1) of a di↵erential equation of Schr¨odinger type 00 n(x)n(x)(x2I+2J)+((2n+ 1)I+2J)n(x)=0 The orthogonal matrix-valued functions (complete in L2(R,CN⇥N)) n(x)=e x2/2Pn(x)eAx are simultaneously eigenfunctions 10 sábado 23 de febrero de 13 THE EXAMPLE Let us consider L2 W(R,CN⇥N) with the weight matrix W(x)=e x2eAxeA⇤x,x2R and the family of MOP (Pn)nsatisfying P00 n(x)+P0 n(x)(2xI+2A)+Pn(x)(A22J)=(2nI2J)Pn(x) 1) of a di↵erential equation of Schr¨odinger type 00 n(x)n(x)(x2I+2J)+((2n+ 1)I+2J)n(x)=0 2) and an integral operator of Fourier type 1 p2⇡ZR n(t)eixtei⇡ 2Jdt =(i)nei⇡ 2Jn(x) The orthogonal matrix-valued functions (complete in L2(R,CN⇥N)) n(x)=e x2/2Pn(x)eAx are simultaneously eigenfunctions 10 sábado 23 de febrero de 13 THE EXAMPLE Let us consider L2 W(R,CN⇥N) with the weight matrix W(x)=e x2eAxeA⇤x,x2R and the family of MOP (Pn)nsatisfying P00 n(x)+P0 n(x)(2xI+2A)+Pn(x)(A22J)=(2nI2J)Pn(x) That means that the second-order di↵erential operator D:L2(R,CN⇥N)!L2(R,CN⇥N) [FD](x)= d2 dx2F(x)F(x)(x2I+2J) and the integral operator F:L2(R,CN⇥N)!L2(R,CN⇥N) [FF](x)= 1 p2⇡ZR F(t)eixtei⇡ 2Jdt commute in L2(R,CN⇥N), i.e. DF=FD 10 sábado 23 de febrero de 13 CONSEQUENCES FOR (Pn)n 11 sábado 23 de febrero de 13 CASE N=2 The normalized family is n(x)=0 @ n(x)/pn+1 ⌫1qn+1 2n+1 n+1(x) ⌫1qn 2n n1(x) n(x)/pn 1 A, n=1+n 2⌫2 1 That means RRn(x)⇤ m(x)dx =nmI2, so the diagonal entries are probability distributions on R. The plots for the first values of nand ⌫1= 1 are given by 13 sábado 23 de febrero de 13 CASE N=2 4 1 3 2 0 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 –4 –2 2 4 t The normalized family is n(x)=0 @ n(x)/pn+1 ⌫1qn+1 2n+1 n+1(x) ⌫1qn 2n n1(x) n(x)/pn 1 A, n=1+n 2⌫2 1 That means RRn(x)⇤ m(x)dx =nmI2, so the diagonal entries are probability distributions on R. The plots for the first values of nand ⌫1= 1 are given by 1 n+1 ( 2 n  2 n+1)+ 2 n+1 13 sábado 23 de febrero de 13 CASE N=2 4 1 3 2 0 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 –4 –2 2 4 t 5 23 0 1 4 0.1 0.2 0.3 0.4 0.5 –4 –2 2 4 t The normalized family is n(x)=0 @ n(x)/pn+1 ⌫1qn+1 2n+1 n+1(x) ⌫1qn 2n n1(x) n(x)/pn 1 A, n=1+n 2⌫2 1 That means RRn(x)⇤ m(x)dx =nmI2, so the diagonal entries are probability distributions on R. The plots for the first values of nand ⌫1= 1 are given by 1 n+1 ( 2 n  2 n+1)+ 2 n+1 1 n( 2 n  2 n1)+ 2 n1 13 sábado 23 de febrero de 13 CASE N=2 4 1 3 2 0 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 –4 –2 2 4 t 5 23 0 1 4 0.1 0.2 0.3 0.4 0.5 –4 –2 2 4 t The normalized family is n(x)=0 @ n(x)/pn+1 ⌫1qn+1 2n+1 n+1(x) ⌫1qn 2n n1(x) n(x)/pn 1 A, n=1+n 2⌫2 1 That means RRn(x)⇤ m(x)dx =nmI2, so the diagonal entries are probability distributions on R. The plots for the first values of nand ⌫1= 1 are given by Physical interpretation? 1 n+1 ( 2 n  2 n+1)+ 2 n+1 1 n( 2 n  2 n1)+ 2 n1 13 sábado 23 de febrero de 13 THREE-TERM RECURRENCE RELATION We also have (xI)nm . =ZR xn(x)⇤ m(x)dx =0 @qnn+1 2n0 0qnn1 2n 1 Am,n1 + 0⌫1 2pnn+1 ⌫1 2pnn+1 0!m,n +0 @q(n+1)n+2 2n+1 0 0q(n+1)n 2n+1 1 Am,n+1 14 sábado 23 de febrero de 13 THREE-TERM RECURRENCE RELATION We also have (xI)nm . =ZR xn(x)⇤ m(x)dx =0 @qnn+1 2n0 0qnn1 2n 1 Am,n1 + 0⌫1 2pnn+1 ⌫1 2pnn+1 0!m,n +0 @q(n+1)n+2 2n+1 0 0q(n+1)n 2n+1 1 Am,n+1 Therefore (xI) is the matrix of the homomorphism F!xF in L2(R,CN⇥N) with respect to the basis (n)n (xI)= 0 B B B B B B B @ 0?? ?00 ? ?00 ?? ?? 00? ?00?? ............... 1 C C C C C C C A 14 sábado 23 de febrero de 13 THREE-TERM RECURRENCE RELATION We also have (xI)nm . =ZR xn(x)⇤ m(x)dx =0 @qnn+1 2n0 0qnn1 2n 1 Am,n1 + 0⌫1 2pnn+1 ⌫1 2pnn+1 0!m,n +0 @q(n+1)n+2 2n+1 0 0q(n+1)n 2n+1 1 Am,n+1 Therefore (xI) is the matrix of the homomorphism F!xF in L2(R,CN⇥N) with respect to the basis (n)n (xI)= 0 B B B B B B B @ 0?? ?00 ? ?00 ?? ?? 00? ?00?? ............... 1 C C C C C C C A Physical interpretation? 14 sábado 23 de febrero de 13 INTEGRAL RELATIONS There will be 8 real integral equations 15 sábado 23 de febrero de 13 INTEGRAL RELATIONS If we call Qn(x)=✓00 01 ◆Pn(x)✓0⌫1x 01 ◆ )ex2/2Q2n(x)= 1 p2⇡ZR et2/2cos(xt)Q2n(t)dt ex2/2Q2n+1(x)=1 p2⇡ZR et2/2sin(xt)Q2n+1(t)dt There will be 8 real integral equations 15 sábado 23 de febrero de 13 INTEGRAL RELATIONS If we call Qn(x)=✓10 00 ◆Pn(x)✓10 00 ◆ )ex2/2Q2n(x)= 1 p2⇡ZR et2/2cos(xt)Q2n(t)dt ex2/2Q2n+1(x)=1 p2⇡ZR et2/2sin(xt)Q2n+1(t)dt There will be 8 real integral equations 15 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) [KnF](x)=ZR F(y)Kn(x, y)dy 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) [KnF](x)=ZR F(y)Kn(x, y)dy IIKS theory 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) 4⇥4 RHP () on C\{[I} [KnF](x)=ZR F(y)Kn(x, y)dy IIKS theory 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) 4⇥4 RHP () on C\{[I} IIKS theory 8 > < > : +()=()(IG()),2⌘[ ()=I+1 +2 2+··· ,!1 G() depends on J2BnJ2 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) 4⇥4 RHP () on C\{[I} IIKS theory 8 > < > : +()=()(IG()),2⌘[ ()=I+1 +2 2+··· ,!1 G() depends on J2BnJ2 ()! () (constant jumps) 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) 4⇥4 RHP () on C\{[I} IIKS theory ()! () (constant jumps) Lax pair @ =A @s =U 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) 4⇥4 RHP () on C\{[I} IIKS theory ()! () (constant jumps) Lax pair @ =A @s =U Compatibility conditions 17 sábado 23 de febrero de 13 NON-COMMUTATIVE PAINLEVÉ IV Kn⇠Kn(x, y) 4⇥4 RHP () on C\{[I} IIKS theory ()! () (constant jumps) Lax pair @ =A @s =U Compatibility conditions ⇢u0=u2+2su+4z2nI2+VA z00 =2u0z+2uz0 2sz0 where VA= 2[J2,y]y1([x,y]=xy yx) and z=(1)0 11,y=2(1)12,u=(1)0 12(1)1 12 +2sI2 17 sábado 23 de febrero de 13